Non-convex sparse regularisation |
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Authors: | Markus Grasmair |
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Institution: | Computational Science Center, University of Vienna, Nordbergstrasse 15, 1090 Vienna, Austria |
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Abstract: | We study the regularising properties of Tikhonov regularisation on the sequence space ?2 with weighted, non-quadratic penalty term acting separately on the coefficients of a given sequence. We derive sufficient conditions for the penalty term that guarantee the well-posedness of the method, and investigate to which extent the same conditions are also necessary. A particular interest of this paper is the application to the solution of operator equations with sparsity constraints. Assuming a linear growth of the penalty term at zero, we prove the sparsity of all regularised solutions. Moreover, we derive a linear convergence rate under the assumptions of even faster growth at zero and a certain injectivity of the operator to be inverted. These results in particular cover non-convex ?p regularisation with 0<p<1. |
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Keywords: | Tikhonov regularisation Sparsity Convergence rates |
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